We extend the classical notion of standardly stratified k-algebra (stated for finite dimensional k-algebras) to the more general class of rings, possibly without 1, with enough idempotents. We show that many of the fundamental results, which are known for classical standardly stratified algebras, can be generalized to this context. Furthermore, new classes of rings appear as: ideally standardly stratified and ideally quasi-hereditary. In the classical theory, it is known that quasi-hereditary and ideally quasi-hereditary algebras are equivalent notions, but in our general setting, this is no longer true. To develop the theory, we use the well-known connection between rings with enough idempotents and skeletally small categories (ringoids or rings with several objects).

In this note, we compute the centers of the categories of tilting modules for G = SL2 in prime characteristic, of tilting modules for the corresponding quantum group at a complex root of unity, and of projective GgT-modules when g = 1, 2.

Given a series we define , by the relations

The series is said to be summable (C, k), where k > - 1, to the sum s if

to be summable (C, - 1) to s if it converges to s and nan = o(l); to be absolutely summable (C, k), or summable | C, k, to s if it is summable (C, k) to s and

and to be strongly Cesàro summable to s with order k > 0 and index p or summable [C; k, p] to s, if

In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis–growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers.

The similarities between martingale convergence theory and pointwise ergodic theory are now well known [5, 7, 9, 10]. In [5] the similarity between the proofs of the Hopf– Dunford–Schwartz individual ergodic theorem and the martingale convergence theorem is systematically exploited to produce very general ” maximal ergodic ” inequalities for certain sequences of contractions on L1-spaces. A different approach by Rota [10] and Rao [9] leads to a unified convergence theory for martingales and Abel limits. Bishop [1] has produced ” upcrossing” inequalities which yield both theChacon-Ornstein theorem [4] and the martingale convergence theorem.

Let T be a dominant operator that is a quasi-affine transform of an M-hyponormal operator. In this paper we show that if f is a function analytic on a neighborhood of the spectrum of T, then Weyl's theorem holds for f(T{\hskip1}).

This paper investigates finite p-groups, p \geq 5, in which every cyclic subgroup has defect at most two. This class of groups is often denoted by {\cal U}_{2,1}. The main result is a theorem which characterises these groups by identifying a family of groups in {\cal U}_{2,1}, and showing that any finite p-group in {\cal U}_{2,1}, with p \geq 5, must be a homomorphic image of one of these groups.

Let G be a group and let ℓ(G) be the set of all conjugacy classes [H] of subgroups H of G, where a partial order ≤ is defined by [H1] ≤ [H2] if and only if H1, is contained in some conjugate of H2.

A number of papers (see for example [1] and the references mentioned there) deal with the question of characterizing groups G by the poset ℓ(G). For example, in [1] it was shown that if ℓ(G) and ℓ(H) are order-isomorphic and G is a noncyclic p-group then |G| = |H|. Moreover, if G is abelian, then G = H, and if G is metacyclic then H is metacyclic.

For algebraic terms which are not defined, one may consult [2]. The symbol S(X) denotes the semigroup, under composition, of all continuous selfmaps of the topological space X. When X is discrete, S(X) is simply the full transformation semigroup on the set X. It has long been known that Green's relations and ℐ coincide for [2, p. 52] and F. A. Cezus has shown in his doctoral dissertation [1, p. 34] that and ℐ also coincide for S(X) when X is the one-point compactification of the countably infinite discrete space. Our main purpose here is to point out the fact that among the 0-dimensional metric spaces, Cezus discovered the only nondiscrete space X with the property that and ℐ coincide on the semigroup S(X). Because of a result in a previous paper [6] by S. Subbiah and the author, this property (for 0-dimensional metric spaces) is in turn equivalent to the semigroup being regular. We gather all this together in the following

Ogasawara and Yoshinaga [9] have shown that a B*-algebra is weakly completely continuous (w.c.c.) if and only if it is *-isomorphic to the B*(∞)-sum of algebras LC(HX), where each LC(HX)is the algebra of all compact linear operators on the Hilbert space Hx. As Kaplansky [5] has shown that a B*-algebra is B*-isomorphic to the B*(∞)-sum of algebras LC(HX) if and only if it is dual, it follows that a 5*-algebra A is w.c.c. if and only if it is dual. We have observed that, if only certain key elements of a B*-algebra A are w.c.c, then A is already dual. This observation constitutes our main theorem which goes as follows. A B*-algebra A is dual if and only if for every maximal modular left ideal M there exists a right identity modulo M that is w.c.c.

By a quasi-permutation matrix we mean a square matrix over the complex field [Copf ] with non-negative integral trace. Thus every permutation matrix over [Copf ] is a quasi-permutation matrix. For a given finite group G, let p(G) denote the minimal degree of a faithful permutation representation of G (or a faithful representation of G by permutation matrices), let q(G) denote the minimal degree of a faithful representation of G by quasi-permutation matrices over the rational field ℚ, and let c(G) be the minimal degree of a faithful representation of G by complex quasi-permutation matrices. See [1].

We study an interesting class of Banach function algebras of infinitely differentiable functions on perfect, compact plane sets. These algebras were introduced by H. G. Dales and A. M. Davie in 1973, called Dales-Davie algebras and denoted by D(X, M), where X is a perfect, compact plane set and M = {M n }∞ n = 0 is a sequence of positive numbers such that M 0 = 1 and (m + n)!/M m+n ≤ (m!/M m )(n!/M n ) for m, n ∈ N. Let d = lim sup(n!/Mn )1/n and Xd = {z ∈ C : dist(z, X) ≤ d}. We show that, under certain conditions on X, every f ∈ D(X, M) has an analytic extension to X d . Let DP [D R ]) be the subalgebra of all f ∈ D(X, M) that can be approximated by the restriction to X of polynomials [rational functions with poles off X]. We show that the maximal ideal space of D P is , the polynomial convex hull of X d , and the maximal ideal space of D R is X d . Using some formulae from combinatorial analysis, we find the maximal ideal space of certain subalgebras of Dales-Davie algebras.

Pointing out the difference between the Discrete Nonlinear Schrödinger equation with the classical power law nonlinearity – for which solutions exist globally, independently of the sign and the degree of the nonlinearity, the size of the initial data and the dimension of the lattice – we prove either global existence or nonexistence, in time, for the Discrete Klein-Gordon equation with the same type of nonlinearity (but of “blow-up” sign), under suitable conditions on the initial data, and sometimes on the dimension of the lattice. The results consider both the conservative and the linearly damped lattice. Similarities and differences with the continuous counterparts are remarked. We also make a short comment on the existence of excitation thresholds, for forced solutions of damped and parametrically driven Klein-Gordon lattices.

Let $A$ be a locally finite, N-graded, noetherian algebra with a balanced dualizing complex. If $A$ is a Hopf algebra, then $A$ has finite injective dimension.

Bai [1] gave an intrinsic integral inequality for compact minimal submanifolds of constant curvature Riemannian manifolds. In this paper, we extend Bai's result to the case of pseudo-umbilical submanifolds.

Let μ be an isomorphism which maps a subgroup A of the group G onto a second subgroup B (not necessarily distinct from A) of G; then μ is called a partial automorphism of G. If A coincides with G, that is if the isomorphism is defined on the whole of G, we speak of a total automorphism; this is what is usually called an automorphism of G. A partial (or total) automorphism μ,* extends or continues a partial automorphism μ if μ* is defined for, at least, all those elements for which μ is defined, and moreover μ* coincides with μ where μ is defined.

We extend the local non-homogeneous Tb theorem of Nazarov, Treil and Volberg to the setting of singular integrals with operator-valued kernel that act on vector-valued functions. Here, ‘vector-valued’ means ‘taking values in a function lattice with the UMD (unconditional martingale differences) property’. A similar extension (but for general UMD spaces rather than UMD lattices) of Nazarov-Treil-Volberg's global non-homogeneous Tb theorem was achieved earlier by the first author, and it has found applications in the work of Mayboroda and Volberg on square-functions and rectifiability. Our local version requires several elaborations of the previous techniques, and raises new questions about the limits of the vector-valued theory.

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

We show that the $\mathbb{Z}$/2-equivariant nth integral Morava K-theory with reality is self-dual with respect to equivariant Anderson duality. In particular, there is a universal coefficients exact sequence in integral Morava K-theory with reality, and we recover the self-duality of the spectrum KO as a corollary. The study of $\mathbb{Z}$/2-equivariant Anderson duality made in this paper gives a nice interpretation of some symmetries of RO($\mathbb{Z}$/2)-graded (i.e. bigraded) equivariant cohomology groups in terms of Mackey functor duality.

In [1] J. Lindenstrauss and M. A. Perles studied the extreme points of the set of all linear operators T of norm ≤ 1 from a finite dimensional Banach space X into itself. In particular they studied the question “When do these extreme points form a semigroup?”.